Timofeeva, N. V. On a new compactification of moduli of vector bundles on a surface. III: Functorial approach. (English. Russian original) Zbl 1230.14060 Sb. Math. 202, No. 3, 413-465 (2011); translation from Mat. Sb. 2011, No. 3, 107-160 (2011). This is the fourth in a series of papers by the author on compactifications of the moduli spaces of vector bundles on surfaces (the first concerned only bundles of rank \(2\) and was unnumbered).Let \(S\) be a smooth irreducible projective surface defined over an algebraically closed field of characteristic zero and let \(L\) be a fixed ample invertible sheaf on \(S\). Let \(M_0\) be the moduli space of stable vector bundles on \(S\) with some fixed Hilbert polynomial. It is well known that \(M_0\) has a natural compactification (the Gieseker-Maruyama compactification) \(\overline{M}\) using S-equivalence classes of semistable torsion-free sheaves on \(S\). In the most recent of the previous papers [Sb. Math. 200, No. 3, 405–427 (2009); translation from Mat. Sb. 200, No. 3, 95–118 (2009; Zbl 1192.14032)], the author completed the construction of a new compactification, denoted in the present paper by \(\widetilde{M}^c\), together with a natural birational morphism \(\widetilde{M}^c\to\overline{M}\), which is an isomorphism over \(M_0\).In the present paper, the author adopts a more functorial approach. For fixed \(S\), \(L\), rank and Hilbert polynomial, she defines a concept of admissible projective scheme \(\widetilde{S}\) with distinguished polarization \(\widetilde{L}\). She then formulates a definition of (semi)stability for pairs \(((\widetilde{S},\widetilde{L}),\widetilde{E})\), where \(\widetilde{E}\) is a locally free sheaf on \(\widetilde{S}\), and uses this to construct a moduli functor. The usual definitions can be carried through to this situation (some care is needed because \(\widetilde{S}\) can have non-reduced components) and it is shown that the moduli functor possesses a coarse moduli scheme \(\widetilde{M}\). Moreover \(\widetilde{M}\) is a projective Noetherian algebraic scheme and there is a birational morphism \(\kappa:\overline{M}\to\widetilde{M}\), which is an isomorphism over the open subset corresponding to stable bundles on \((S,L)\). There is also a relation of M-equivalence such that two semistable pairs are M-equivalent if and only if they correspond to the same point of \(\widetilde{M}\). Reviewer: P. E. Newstead (Liverpool) Cited in 4 ReviewsCited in 6 Documents MSC: 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14D20 Algebraic moduli problems, moduli of vector bundles 14M27 Compactifications; symmetric and spherical varieties Keywords:Moduli space; vector bundle; semistable coherent sheaf; moduli functor; algebraic surface Citations:Zbl 1192.14032 PDFBibTeX XMLCite \textit{N. V. Timofeeva}, Sb. Math. 202, No. 3, 413--465 (2011; Zbl 1230.14060); translation from Mat. Sb. 2011, No. 3, 107--160 (2011) Full Text: DOI arXiv